A machine fills milk into $500$ml packages. It is suspected that the machine is not working correctly and that the amount of milk filled differs from the setpoint $\mu_0 = 500$. A sample of $226$ packages filled by the machine are collected. The sample mean $\bar{y}$ is equal to $517.2$ and the sample variance $s^2_{n-1}$ is equal to $262.56$.
Test the hypothesis that the amount filled corresponds on average to the setpoint. What is the value of the t-test statistic?
The t-test statistic is calculated by: $\begin{aligned} t & = & \frac{\bar y - \mu_0}{\sqrt{\frac{s^2_{n-1}}{n}}} = \frac{517.2 - 500}{\sqrt{\frac{262.56}{226}}} = 15.958. \end{aligned}$ The t-test statistic is thus equal to $15.958$.